Most likely not. Although you and I are able to set aside our different views and work together, asking both factions as an entireity to do so seems highly improbable. While we may not hold each other at gunpoint someday, it may be quite some time before full cooperation between our Factions could ever come close to being achieved.

That is unfortunate, and should the New Monarchy claim the power they wish to possess what will keep the peace if our factions will not cooperate now?

We will keep the peace. Your faction will go on to pursue its own interests as it has. Dead Orbit will be... Hmm... More complicated... Either way, I am not saying our Factions will fight with each other. I am saying that we will most likely niether help or hinder one another. It is likely that we can get along, but unlikely that we will actually help each other in any major way. Unless of course, the City should fall under seige. We have already agreed to work together if the situation is dire enough, have we not? We will all work together to defend our home. (Except maybe Dead Orbit)

I was more worried that New Monarchy would look on the War Cult as a belligerent war-mongering faction that would need to be removed. We may be called that by the ignorant but are you sure New Monarchy would not take the opportunity to eliminate the War Cult from the equation entirely?

Read their 7 tenets. NM only encourages "cooperation" between factions when that means "I'm in charge and you all listen"